Probability Abstracts 64
This document contains abstracts 1761-1798.
They have been mailed on August 31, 2001.
1761. INFORMATION FLOW ON TREES
Elchanan Mossel and Yuval Peres
Consider a tree network $T$, where each edge acts as an independent copy of a
given channel $M$, and information is propagated from the root. For which $T$
and $M$ does the configuration obtained at level $n$ of $T$ typically contain
significant information on the root variable? This problem arose independently
in biology, information theory and statistical physics.
For all $b$, we construct a channel for which the variable at the root of the
$b$-ary tree is independent of the configuration at level 2 of that tree, yet
for sufficiently large $B>b$, the mutual information between the configuration
at level $n$ of the $B$-ary tree and the root variable is bounded away from
zero. This is related to certain secret-sharing protocols.
We improve the upper bounds on information flow for asymmetric binary
channels (which correspond to the Ising model with an external field) and for
symmetric $q$-ary channels (which correspond to Potts models).
Let $\lam_2(M)$ denote the second largest eigenvalue of $M$, in absolute
value. A CLT of Kesten and Stigum~(1966) implies that if $b |\lam_2(M)|^2 >1$,
then the {\em census} of the variables at any level of the $b$-ary tree,
contains significant information on the root variable. We establish a converse:
if $b |\lam_2(M)|^2 < 1$, then the census of the variables at level $n$ of the
$b$-ary tree is asymptotically independent of the root variable. This contrasts
with examples where $b |\lam_2(M)|^2 <1$, yet the {\em configuration} at level
$n$ is not asymptotically independent of the root variable.
mossel@microsoft.com
- This article is available
from the xxx mathematics archive as
math.PR/0107033 .
1762. MARKOV CHAIN INTERSECTIONS AND THE LOOP-ERASED WALK
Russell Lyons, Yuval Peres and Oded Schramm
Let X and Y be independent transient Markov chains on the same state space
that have the same transition probabilities. Let L denote the ``loop-erased
path'' obtained from the path of X by erasing cycles when they are created. We
prove that if the paths of X and Y have infinitely many intersections a.s.,
then L and Y also have infinitely many intersections a.s.
schramm@microsoft.com
- This article is available
from the xxx mathematics archive as
math.PR/0107055 .
1763. RANDOM ELECTRICAL NETWORKS ON COMPLETE GRAPHS II: PROOFS
Geoffrey Grimmett and Harry Kesten
This paper contains the proofs of Theorems 2 and 3 of the article entitled
Random Electrical Networks on Complete Graphs, written by the same authors and
published in the Journal of the London Mathematical Society, vol. 30 (1984),
pp. 171-192. The current paper was written in 1983 but was not published in a
journal, although its existence was announced in the LMS paper. This TeX
version was created on 9 July 2001. It incorporates minor improvements to
formatting and punctuation, but no change has been made to the mathematics.
We study the effective electrical resistance of the complete graph $K_{n+2}$
when each edge is allocated a random resistance. These resistances are assumed
independent with distribution $P(R=\infty)=1-n^{-1}\gamma(n)$, $P(R\le x) =
n^{-1}\gamma(n)F(x)$ for $0\le x < \infty$, where $F$ is a fixed distribution
function and $\gamma(n)\to\gamma\ge 0$ as $n\to\infty$. The asymptotic
effective resistance between two chosen vertices is identified in the two cases
$\gamma\le 1$ and $\gamma>1$, and the case $\gamma=\infty$ is considered. The
analysis proceeds via detailed estimates based on the theory of branching
processes.
g.r.grimmett@statslab.cam.ac.uk
- This article is available
from the xxx mathematics archive as
math.PR/0107068 .
1764. RIEMANN-HILBERT PROBLEMS FOR LAST PASSAGE PERCOLATION
Jinho Baik
Last three years have seen new developments in the theory of last passage
percolation, which has variety applications to random permutations, random
growth and random vicious walks. It turns out that a few class of models have
determinant formulas for the probability distribution, which can be analyzed
asymptotically. One of the tools for the asymptotic analysis has been the
Riemann-Hilbert method. In this paper, we survey the use of Riemann-Hilbert
method in the last passage percolation problems.
jbaik@math.princeton.edu
- This article is available
from the xxx mathematics archive as
math.PR/0107079 .
1765. VARIATIONAL PRINCIPLE AND ALMOST QUASILOCALITY FOR SOME RENORMALIZED
MEASURES
Roberto Fernandez, Arnaud Le Ny, Frank Redig
We restore part of the thermodynamic formalism for some renormalized measures
that are known to be non-Gibbsian. We first point out that a recent theory due
to Pfister implies that for block-transformed measures free energies and
relative entropy densities exist and are conjugate convex functionals. We then
determine a necessary and sufficient condition for consistency with a
specification that is quasilocal in a fixed direction. As corollaries we obtain
consistency results for models with FKG monotonicity and for models with
appropriate "continuity rates". For (noisy) decimations or projections of the
Ising model, these results imply almost quasilocality of the decimated "+" and
"-" measures.
aleny@maths.univ-rennes1.fr
- This article is available
from the xxx mathematics archive as
math.PR/0107081 .
1766. A PERCOLATION FORMULA
Oded Schramm
Let $A$ be an arc on the boundary of the unit disk $U$. We prove an
asymptotic formula for the probability that there is a percolation cluster $K$
for critical site percolation on the triangular grid in $U$ which intersects
$A$ and such that 0 is surrounded by $K\cup A$.
schramm@microsoft.com
- This article is available
from the xxx mathematics archive as
math.PR/0107096 .
1767. AN EXTENSION TO THE TANGENT SEQUENCE MARTINGALE INEQUALITY
Stephen Montgomery-Smith and Shih-Chi Shen
For each 1 < p < infinity, there exists a positive constant c_p, depending
only on p, such that the following holds. Let (d_k), (e_k) be real-valued
martingale difference sequences. If for for all bounded nonnegative predictable
sequences (s_k) and all positive integers k we have E[s_k vee |e_k|] le E[s_k
vee |d_k|] then for all positive integers n we have || sum_{k=1}^n e_k ||_p le
c_p || \sum_{k=1}^n d_k ||_p .
stephen@math.missouri.edu
- This article is available
from the xxx mathematics archive as
math.PR/0107120 .
1768. FILTRATIONS OF RANDOM PROCESSES IN THE LIGHT OF CLASSIFICATION THEORY.
I. A TOPOLOGICAL ZERO-ONE LAW
Boris Tsirelson
Filtered probability spaces (called "filtrations" for short) are shown to
satisfy such a topological zero-one law: for every property of filtrations,
either the property holds for almost all filtrations, or its negation does. In
particular, almost all filtrations are conditionally nonatomic.
An accurate formulation is given in terms of orbit equivalence relations on
Polish G-spaces. The set of all isomorphic classes of filtrations may be
identified with the orbit space X/G for a special Polish G-space X. A "property
of filtrations" means a G-invariant subset of X having the Baire property.
"Almost all filtrations" means a comeager subset of X (the Baire category
approach). The zero-one law is a kind of ergodicity of X. It holds for
filtrations both in discrete and continuous time.
The interplay between probability theory and descriptive set theory could be
interesting for both parties.
tsirel@math.tau.ac.il
- This article is available
from the xxx mathematics archive as
math.PR/0107121 .
1769. NONPARAMETRIC VOLATILITY DENSITY ESTIMATION
Bert van Es, Peter Spreij and Harry van Zanten
We consider two kinds of stochastic volatility models. Both kinds of models
contain a stationary volatility process, the density of which, at a fixed
instant in time, we aim to estimate.
We discuss discrete time models where for instance a log price process is
modeled as the product of a volatility process and i.i.d. noise. We also
consider samples of certain continuous time diffusion processes. The sampled
time instants will be be equidistant with vanishing distance.
A Fourier type deconvolution kernel density estimator based on the logarithm
of the squared processes is proposed to estimate the volatility density.
Expansions of the bias and bounds on the variances are derived.
spreij@science.uva.nl
- This article is available
from the xxx mathematics archive as
math.PR/0107135 .
1770. GEOMETRY OF THE UNIFORM SPANNING FOREST:
TRANSITIONS IN DIMENSIONS 4, 8, 12
Itai Benjamini, Harry Kesten, Yuval Peres, Oded Schramm
The uniform spanning forest (USF) in $Z^d$ is the weak limit of random,
uniformly chosen, spanning trees in $[-n,n]^d$. Pemantle proved that the USF
consists a.s. of a single tree if and only if $d \le 4$. We prove that any two
components of the USF in $Z^d$ are adjacent a.s. if $5 \le d \le 8$, but not if
$d \ge 9$. More generally, let $N(x,y)$ be the minimum number of edges outside
the USF in a path joining $x$ and $y$ in $Z^d$. Then a.s. $\max\{N(x,y) :
x,y\in Z^d\}$ is the integer part of $(d-1)/4$. The notion of stochastic
dimension for random relations in the lattice is introduced and used in the
proof.
schramm@microsoft.com
- This article is available
from the xxx mathematics archive as
math.PR/0107140 .
1771. ON HIDDEN MARKOV CHAINS AND FINITE STOCHASTIC SYSTEMS
Peter Spreij
In this paper we study various properties of finite stochastic systems or
hidden Markov chains as they are alternatively called. We discuss their
construction following different approaches and we also derive recursive
filtering formulas for the different systems that we consider. The key tool is
a simple lemma on conditional expectations.
spreij@science.uva.nl
- This article is available
from the xxx mathematics archive as
math.PR/0107144 .
1772. WEAK LAWS IN GEOMETRIC PROBABILITY
Mathew D. Penrose and J. E. Yukich
Using a coupling argument, we establish a general weak law of large numbers
for functionals of binomial point processes in d-dimensional space, with a
limit that depends explicitly on the (possibly non-uniform) density of the
point process. The general result is applied to the minimal spanning tree, the
k-nearest neighbors graph, the Voronoi graph, and the sphere of influence
graph. Functionals of interest include total edge length with arbitrary
weighting, number of vertices of specifed degree, and number of components. We
also obtain weak laws for functionals of marked point processes, including
statistics of Boolean models.
mathew.penrose@durham.ac.uk
- This article is available
from the xxx mathematics archive as
math.PR/0107149 .
1773. SAMPLE PATH PROPERTIES AND RECURRENCE OF A STABLE-LIKE PROCESS OVER AN
INFINITE EXTENSION OF A LOCAL FIELD
Anatoly N. Kochubei
We consider an infinite extension $K$ of a local field of zero characteristic
which is a union of an increasing sequence of finite extensions. $K$ is
equipped with an inductive limit topology; its conjugate $\overline{K}$ is a
completion of $K$ with respect to a topology given by certain explicitly
written seminorms. The semigroup of measures, which defines a stable-like
process $X(t)$ on $\overline{K}$, is concentrated on a compact subgroup
$S\subset \overline{K}$. We study properties of the process $X_S(t)$, a part of
$X(t)$ in $S$. It is shown that $X_S(t)$ is recurrent, the Hausdorff and
packing dimensions of the image of an interval equal 0 almost surely. In the
case of tamely ramified extensions a correct Hausdorff measure for this set is
found.
ank@ank.kiev.ua
- This article is available
from the xxx mathematics archive as
math.PR/0107156 .
1774. EXISTENCE OF GIBBS MEASURES RELATIVE TO BROWNIAN MOTION
Volker Betz
We prove existence of infinite volume Gibbs measures relative to Brownian
motion. We require the pair potential W to fulfill a uniform integrability
condition, but otherwise our restrictions on the potentials are relatively
weak. In particular, our results are applicable to the massless Nelson model.
We also prove an upper bound for path fluctuations under the infinite volume
Gibbs measures.
betz@mathematik.tu-muenchen.de
- This article is available
from the xxx mathematics archive as
math.PR/0107184 .
1775. COVER TIMES FOR BROWNIAN MOTION AND RANDOM WALKS IN TWO DIMENSIONS
Amir Dembo, Yuval Peres, Jay Rosen, Ofer Zeitouni
Let T(x,r) denote the first hitting time of the disc of radius r centered at
x for Brownian motion on the two dimensional torus. We prove that sup_{x}
T(x,r)/|log r|^2 --> 2/pi as r --> 0. The same applies to Brownian motion on
any smooth, compact connected, two-dimensional, Riemannian manifold with unit
area and no boundary. As a consequence, we prove a conjecture, due to Aldous
(1989), that the number of steps it takes a simple random walk to cover all
points of the lattice torus Z_n^2 is asymptotic to (2n log n)^2/pi. Determining
these asymptotics is an essential step toward analyzing the fractal structure
of the set of uncovered sites before coverage is complete; so far, this
structure was only studied non-rigorously in the physics literature. We also
establish a conjecture, due to Kesten and Revesz, that describes the
asymptotics for the number of steps needed by simple random walk in Z^2 to
cover the disc of radius n.
amir@stat.stanford.edu
- This article is available
from the xxx mathematics archive as
math.PR/0107191 .
1776. DIRECTED PERCOLATION AND RANDOM WALK
Geoffrey Grimmett and Philipp Hiemer
Techniques of `dynamic renormalization', developed earlier for undirected
percolation and the contact model, are adapted to the setting of directed
percolation, thereby obtaining solutions of several problems for directed
percolation on $Z^d$ where $d \ge 2$. The first new result is a type of
uniqueness theorem: for every pair $x$ and $y$ of vertices which lie in
infinite open paths, there exists almost surely a third vertex $z$ which is
joined to infinity and which is attainable from $x$ and $y$ along directed open
paths. Secondly, it is proved that a random walk on an infinite directed
cluster is transient, almost surely, when $d \ge 3$. And finally, the block
arguments of the paper may be adapted to systems with infinite range, subject
to certain conditions on the edge probabilities.
g.r.grimmett@statslab.cam.ac.uk
- This article is available
from the xxx mathematics archive as
math.PR/0108062 .
1777. PROOF OF THE CONJECTURE THAT THE PLANAR SELF-AVOIDING WALK HAS ROOT MEAN
SQUARE DISPLACEMENT EXPONENT 3/4
Irene Hueter
This paper proves the long-standing open conjecture rooted in chemical
physics (Flory (1949)) that the self-avoiding walk (SAW) in the square lattice
has root mean square displacement exponent \nu= 3/4. This value is an instance
of the formula \nu=1 on Z and \nu = max(1/2, 1/4 + 1/d) in Z^d for dimensions d
\geq 2, which will be proved in a subsequent paper. This expression differs
from the one that Flory's arguments suggested. We consider (a) the point
process of self-intersections defined via certain paths of the symmetric simple
random walk in Z^2 and (b) a ``weakly self-avoiding cone process'' relative to
this point process when in a certain "shape". We derive results on the
asymptotic expected distance of the weakly SAW with parameter \beta>0 from its
starting point, from which a number of distance exponents are immediately
collectable for the SAW as well. Our method employs the Palm distribution of
the point process of self-intersection points in a cone.
hueter@math.ufl.edu
- This article is available
from the xxx mathematics archive as
math.PR/0108077 .
1778. BUILDING A STATIONARY STOCHASTIC PROCESS FROM A FINITE-DIMENSIONAL
MARGINAL
Marcus Pivato
If A is a finite alphabet, Z^D is a D-dimensional lattice, U is a subset of
Z^D, and mu_U is a probability measure on A^U that ``looks like'' the marginal
projection of a stationary random field on A^(Z^D), then can we ``extend'' mu_U
to such a field? Under what conditions can we make this extension ergodic,
(quasi)periodic, or (weakly) mixing? After surveying classical work on this
problem when D = 1, we provide some sufficient conditions and some necessary
conditions for mu_U to be extendible for D > 1, and show that, in general, the
problem is not formally decidable.
pivato@math.toronto.edu
- This article is available
from the xxx mathematics archive as
math.PR/0108081 .
1779. SHUFFLES ON COXETER GROUPS
Swapneel Mahajan
The random-to-top and the riffle shuffle are two well-studied methods for
shuffling a deck of cards. These correspond to the symmetric group $S_n$, i.e.,
the Coxeter group of type $A_{n-1}$. In this paper, we give analogous shuffles
for the Coxeter groups of type $B_n$ and $D_n$. These can be interpreted as
shuffles on a ``signed'' deck of cards. With these examples as motivation, we
abstract the notion of a shuffle algebra which captures the connection between
the algebraic structure of the shuffles and the geometry of the Coxeter groups.
We also briefly discuss the generalisation to buildings which leads to
q-analogues.
swapneel@math.cornell.edu
- This article is available
from the xxx mathematics archive as
math.PR/0108094 .
1780. FORMULA FOR THE MEAN SQUARE DISPLACEMENT EXPONENT OF THE SELF-AVOIDING
WALK IN 3, 4 AND ALL DIMENSIONS
Irene Hueter
This paper proves the formula \nu(d) =1 for d=1 and \nu(d) = max(1/4 +1/d,
1/2) for d > 1 for the root mean square displacement exponent \nu(d) of the
self-avoiding walk (SAW) in Z^d, and thus, resolves some major long-standing
open conjectures rooted in chemical physics (Flory, 1949). The values \nu(2)
=3/4 and \nu(4) = 1/2 coincide with those that were believed on the basis of
heuristic and "numerical evidence". Perhaps surprisingly, there was no precise
conjecture in dimension 3. Yet as early as in the 1980ies, Monte Carlo
simulations produced a couple of confidence intervals for the exponent \nu(3).
This work is a follow-up to Hueter (2001), which proves the result for d=2 and
lays out the fundamental building blocks for the analysis in all dimensions. We
consider (a) the point process of self-intersections defined via certain paths
of length n of the symmetric simple random walk in Z^d and (b) a ``weakly
self-avoiding cone process'' relative to this point process in a certain
"shape". The asymptotic expected distance of the process in (b) can be
calculated rather precisely as n tends large and, if the point process has
circular shape, can be shown to asymptotically equal (up to error terms) the
one of the weakly SAW with parameter \beta >0. From these results, a number of
distance exponents are immediately collectable for the SAW as well. Our
approach invokes the Palm distribution of the point process of
self-intersections in a cone.
hueter@math.ufl.edu
- This article is available
from the xxx mathematics archive as
math.PR/0108120 .
1781. TRANSIENCE OF SECOND-CLASS PARTICLES AND DIFFUSIVE BOUNDS FOR ADDITIVE
FUNCTIONALS IN ONE-DIMENSIONAL ASYMMETRIC EXCLUSION PROCESSES
Timo Seppalainen and Sunder Sethuraman
Consider a one-dimensional exclusion process with finite-range
translation-invariant jump rates with non-zero drift. Let the process be
stationary with product Bernoulli invariant distribution at density \rho. Place
a second class particle initially at the origin. For the case \rho different
from 1/2 we show that the time spent by the second class particle at the origin
has finite expectation. This strong transience is then used to prove that
variances of additive functionals of local mean-zero functions are diffusive
when \rho is not 1/2. As a corollary to previous work, we deduce the invariance
principle for these functionals. The main arguments are comparisons of H_{-1}
norms, a large deviation estimate for second-class particles, and a relation
between occupation times of second-class particles and additive functional
variances.
seppalai@math.wisc.edu
- This article is available
from the xxx mathematics archive as
math.PR/0108171 .
1782. DIFFUSIVE FLUCTUATIONS FOR ONE-DIMENSIONAL TOTALLY ASYMMETRIC
INTERACTING RANDOM DYNAMICS
Timo Seppalainen
We study central limit theorems for a totally asymmetric, one-dimensional
interacting random system. The models we work with are the
Aldous-Diaconis-Hammersley process and the related stick model. The A-D-H
process represents a particle configuration on the line, or a 1-dimensional
interface on the plane which moves in one fixed direction through random local
jumps. The stick model is the process of local slopes of the A-D-H process, and
has a conserved quantity. The results describe the fluctuations of these
systems around the deterministic evolution to which the random system converges
under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean
fluctuations on the scale of the classical central limit theorem. In the
scaling limit these fluctuations obey deterministic equations with random
initial conditions given by the initial fluctuations. Of particular interest is
the effect of macroscopic shocks, which play a dominant role because dynamical
noise is suppressed on the scale we are working.
seppalai@math.wisc.edu
- This article is available
from the xxx mathematics archive as
math.PR/0108174 .
1783. ONE-ARM EXPONENT FOR CRITICAL 2D PERCOLATION
Gregory F. Lawler, Oded Schramm, Wendelin Werner
The probability that the cluster of the origin in critical site percolation
on the triangular grid has diameter larger than $R$ is proved to decay like
$R^{-5/48}$ as $R\to\infty$.
schramm@microsoft.com
- This article is available
from the xxx mathematics archive as
math.PR/0108211 .
1784. GAUSSIAN LIMITS ASSOCIATED WITH THE POISSON-DIRICHLET
DISTRIBUTION AND THE EWENS SAMPLING FORMULA
Paul Joyce, Stephen M. Krone and Thomas G. Kurtz
In this paper we consider large $\theta$ approximations for
the stationary distribution of the neutral infinite alleles
model as described by the the Poisson--Dirichlet
distribution with parameter $\theta$. We prove a variety of
Gaussian limit theorems for functions of the population
frequencies as the mutation rate $\theta$ goes to infinity.
In particular, we show that if a sample of size $n$ is
drawn from a population described by the Poisson--Dirichlet
distribution, then the conditional probability of a
particular sample configuration is asymptotically normal
with mean and variance dertermined by the Ewens sampling
formula. The asymptotic normality of the conditional
sampling distribution is somewhat surprising since it is a
fairly complicated function of the population frequencies.
Along the way, we also prove an invariance principle giving
weak convergence at the process level for powers of the
size-biased allele frequencies.
joyce@uidaho.edu krone@uidaho.edu kurtz@math.wisc.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
1785. COALESCENT THEORY FOR SEED BANK MODELS
Ingemar Kaj, Stephen M. Krone and Martin Lascoux
We study the genealogical structure of samples from a
population for which any given generation is made up of
direct descendents from several previous generations.
These occur in nature when there are seed banks or egg
banks allowing an individual to leave offspring several
generations in the future. We show how this temporal
structure in the reproduction mechanism causes a decrease
in the coalescence rate. We also investigate the effects
of age-dependent neutral mutations. Our main result gives
weak convergence of the scaled ancestral process, with
the usual diffusion scaling, to a coalescent process which
is equivalent to a time-changed version of Kingman's
coalescent.
ikaj@math.uu.se krone@uidaho.edu Martin.Lascoux@ebc.uu.se
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1786. DISORDER IN PHYSICAL SYSTEMS,
A VOLUME IN HONOUR OF JOHN M. HAMMERSLEY
Geoffrey Grimmett, Dominic Welsh (eds)
First published in 1990 by Oxford University
Press, this volume is now republished electronically
at the url given below. It includes the following
papers:
David Kendall: Speech proposing the toast to John
Hammersley - 1 October 1987.
N. H. Bingham and U. Stadtmüller: Jakimovski methods
and almost-sure convergence.
Peter Clifford: Markov random fields in statistics.
Cyril Domb: On Hammersley's method for one-dimensional
covering problems.
P. Erdös and A. Sárközy: On a problem of Straus.
J.W. Essam and D. Tanlakishani: Directed compact
percolation II: Nodal Points, mass distribution,
and scaling.
Michael E. Fisher and Rajiv R.P. Singh: Critical
points, large-dimensionality expansions, and
the Ising spin glass.
R.J. Gibbens, P.J. Hunt, and F.P. Kelly: Bistability
in communication networks.
I.J. Good: A quantal hypothesis for hadrons and the
judging of physical numerology.
G.R. Grimmett and C.M. Newman: Percolation in
$\infty+1$ dimensions.
D.C. Handscomb: Monte Carlo methods applied to
quantum-mechanical order-disorder phenomena in crystals.
Wilfrid S. Kendall: The diffusion of Euclidean shape.
Harry Kesten: Asymptotics in high dimensions for
percolation.
J.F.C. Kingman: Some random collections of finite
subsets.
C.J.H. McDiarmid: Probabilistic analysis of tree
search.
J.S. Rowlinson: Probability densities for some
one-dimensional problems in statistical mechanics.
J. Michael Steele: Seedlings in the theory of shortest
paths.
D.J.A. Welsh: The computational complexity of some
classical problems from statistical physics.
S.G. Whittington and C.E. Soteros: Lattice animals:
Rigorous results and wild guesses.
P. Whittle: Fields and flows on random graphs.
John C. Wierman: Bond percolation critical probability
bounds for the Kagomé lattice by a substitution method.
David Williams: Brownian motion and the Riemann
zeta-function.
grg@statslab.cam.ac.uk
- To see a preprint or other
information provided by the author
click here.
1787. LARGE DEVIATIONS FOR A MARKOV CHAIN IN A RANDOM LANDSCAPE
Nadine Guillotin-Plantard
Let $(S_{n})_{n\geq 0}$ be a Markov chain defined on
a probability space $(\Omega, {\cF}, P)$ taking values
in a countable state space $E$. Let $(\xi_{x})_{x\in E}$
be a sequence of $\bbR^{p}$-valued random variables
defined on another probability space $(E, {\cA}, \mu)$.
These random variables play the r\^{o}le of the
random landscape, also called random scenery. We
study the large (and moderate) deviations of
the 'cumulative sum'
$Z_{n}=\sum_{k=0}^{n}\xi_{S_{k}}.$
Two kinds of results are obtained :
1- Large deviations principles for the sum $Z_{n}$
with respect to the product measure $P \otimes\mu$
in the case when a large deviations principle is
already known for the occupation measure of
the Markov chain.
2- Large deviations principles for the sum $Z_{n}$
conditionally to the realization of the Markov chain
in the case when the state space is equal to $Z^{d}$
and $(S_{n})_{n\geq 0}$ is a random walk.
guilloti@jonas.univ-lyon1.fr
1788. A NON-STOPPING TIME WITH AN OPTIONAL-STOPPING PROPERTY
David Williams
This is in response to a question asked by Chris Rogers.
He and I hope to fit this example into a sensible picture of
when optional-stopping theorems hold.
Let $B$ be canonical Brownian motion starting at $0$. Define
$\tau$ to be the hitting time of $1$,
$\sigma$ to be the time of the last visit to $0$
before $\tau$,
$\rho$ to be the time at which $B$ attains its maximum value
on $[0,\sigma]$,
all exactly as in Lemma VI.23.1 of the book by
Rogers and Williams.
Then, it is easily deduced from the equality of the optional
and dual previsible projections of $I_{[\rho,\infty)} that
for any bounded (or $L^2$-bounded) martingale $M$, we
have
$E(M_{\rho}) = E(M_0)$.
(That's the paper, not just the abstract!)
dswilliams@markov.fsnet.co.uk
1789. GIRSANOV AND FEYNMAN-KAC TYPE TRANSFORMATIONS
FOR SYMMETRIC MARKOV PROCESSES
Zhen-Qing Chen and Tu-Sheng Zhang
Studied in this paper is the transformation of an
arbitrary symmetric Markov process $X$ by
multiplicative functionals which are the
exponential of continuous additive functionals
of $X$ having zero quadratic variations.
We characterize the transformed semigroups by
their associated quadratic forms.
This is done by first identifying the
symmetric Markov process under Girsanov transform,
which may be of independent interest, and then
applying Feynman-Kac transform to the
Girsanov transformed process.
Stochastic analysis for discontinuous martingales
is used in our approach.
zchen@math.washington.edu tzhang@maths.man.ac.uk
- To see a preprint or other
information provided by the author
click here.
1790. GENERAL GAUGE AND CONDITIONAL GAUGE THEOREMS
Zhen-Qing Chen and Renming Song
General gauge and conditional gauge theorems
are established for a large class of (not necessary
symmetric) strong Markov processes, including Brownian
motions with singular drifts and symmetric stable
processes. Furthermore new classes of functions are
introduced under which the general gauge and conditional
gauge theorems hold. These classes are larger than the
classical Kato class when the process is Brownian motion
in a bounded $C^{1, 1}$ domain.
zchen@math.washington.edu rsong@math.uiuc.edu
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here.
1791. CONDITIONAL GAUGE THEOREM FOR DISCONTINUOUS
ADDITIVE FUNCTIONALS
Zhen-Qing Chen and Renming Song
General gauge and conditional gauge theorems
for discontinuous additive functionals of
Markov processes are given. The generator of the
Schrodinger semigroup given by discontinuous
additive functional is characterized in terms of
its associated bilinear form.
zchen@math.washington.edu rsong@math.uiuc.edu
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information provided by the author
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here.
1792. STRETCHED EXPONENTIAL FIXATION IN STOCHASTIC ISING
MODELS AT ZERO TEMPERATURE
L. R. Fontes, R. H. Schonmann and V. Sidoravicius
We study a class of continuous time Markov processes, which describes
$\pm 1$ spin flip dynamics on the hypercubic lattice $Z^d, d \ge 2$,
with initial spin configurations chosen according to the Bernoulli product
measure with density $p$ of spins $+1$.
During the evolution the spin at each site flips at rate $c = 0,$ or
$0< \alpha \le 1$, or $1$, depending on whether, respectively, a majority
of spins of nearest neighbors to this site exists and agrees with the
value of the spin at the given site, or does not exist (there is a tie),
or exists and disagrees with the value of the spin at the given site.
These dynamics correspond to various stochastic Ising models at 0
temperature, for the Hamiltonian with uniform ferromagnetic interaction
between nearest neighbors. In case $\alpha =1$, the dynamics is also a
threshold voter model. We show that if $p$ is sufficiently
close to $1$, then the system fixates in the sense that for almost every
realization of the initial configuration and dynamical evolution, each site
flips only finitely many times, reaching eventually the state $+1$.
Moreover, we show that in this case the probability $q(t)$ that a given
spin is in state $-1$ at time $t$ satisfies the bound: for arbitrary
$\epsilon > 0$, $q(t) \le \exp(-t^{(1/d) - \epsilon})$, for large $t$.
In $d=2$ we obtain the complementary bound: for arbitrary
$\epsilon > 0$, $q(t) \ge \exp(-t^{(1/2) + \epsilon})$, for large $t$.
lrenato@ime.usp.br rhs@math.ucla.edu vladas@impa.br
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1793. WEAK UNIMODALITY OF FINITE MEASURES, AND AN APPLICATION TO
POTENTIAL THEORY OF ADDITIVE LEVY PROCESSES
Davar Khoshnevisan and Yimin Xiao
A probability measure $\mu$ on $\R^d$ is called `weakly unimodal'
if there exists a constant $\kappa \ge 1$ such that for all $r>0$,
$sup_{a\in\R^d} \mu(B(a, r)) \le \kappa \mu(B(0, r)).$
Here, $B(a, r)$ denotes the
$\ell^\infty$-ball centered at $a\in\R^d$ with radius $r>0$.
In this note, we derive a sufficient condition for weak unimodality
of a measure on the Borel subsets of $\R^d$. In particular, we use
this to prove that every symmetric infinitely divisible
distribution is weakly unimodal. This result is then applied to
improve some of recent results of the authors on capacities and level
sets of additive L\'evy processes.
davar@math.utah.edu
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information provided by the author
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1794. EIGENVALUES OF RANDOM WREATH PRODUCTS
Steven N. Evans
Consider a uniformly chosen element $X_n$ of the $n$-fold
wreath product $\Gamma_n = G \wr G \wr \dots \wr G$,
where $G$ is a finite permutation group acting transitively
on some set of size $s$. The eigenvalues of $X_n$ in the
natural $s^n$-dimensional permutation representation are
investigated by considering the random measure $\Xi_n$
on the unit circle that assigns mass $1$ to each eigenvalue.
It is shown that if $f$ is a trigonometric polynomial, then
$\lim_{n \rightarrow \infty}
P\{\int f d\Xi_n \ne s^n \int f d\lambda\}=0$,
where $\lambda$ is normalised Lebesgue measure on the
unit circle. In particular, $s^{-n} \Xi_n$ converges weakly
in probability to
$\lambda$ as $n \rightarrow \infty$.
For a large class of test functions $f$ with non-terminating
Fourier expansions, it is shown that there exists a constant
$c$ and an almost surely non-zero random variable $W$
(both depending on $f$) such that $c^{-n} \int f d\Xi_n$
converges in distribution as $n \rightarrow \infty$ to $W$.
These results have applications to Sylow $p$-groups of
symmetric groups and autmorphism groups of regular
rooted trees.
evans@stat.berkeley.edu
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information provided by the author
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here.
1795. GEOMETRIC ANALYSIS FOR SYMMETRIC FLEMING-VIOT OPERATORS:
RADEMACHER'S THEOREM AND EXPONENTIAL FAMILIES
Alexander Schied
We use the natural geometry of a symmetric Fleming-Viot
operator $\cL$ to obtain analytical descriptions of
the corresponding Dirichlet space $(\cE,D(\cE))$. In
particular, we give a complete characterization of
functions in $D(\cE)$ in terms of their differentiability
properties along exponential families. Moreover, we prove
a Rademacher theorem stating that any function which is
Lipschitz continuous with respect to the Bhattacharya
distance is contained in $D(\cE)$ and possesses a bounded
gradient. A converse to this statement is also given. Thus,
we relate the Bhattacharya distance to the potential theory
of $\cL$.
schied@mathematik.hu-berlin.de
1796. CHAOTIC SIZE DEPENDENCE IN THE ISING MODEL WITH RANDOM
BOUNDARY CONDITIONS
Aernout C. D. van Enter, Igor Medved and Karel Netocny
We study the nearest-neighbour Ising model with a class of
random boundary conditions, chosen from a symmetric i.i.d.
distribution. We show for dimensions 4 and higher that
almost surely the only limit points for a sequence of
increasing cubes are the plus and the minus state. For d=2
and d=3 we prove a similar result for sparse sequences of
increasing cubes. This question was raised by Newman and
Stein. Our results imply that the Newman-Stein metastate is
concentrated on the plus and the minus state.
aenter@phys.rug.nl I.Medved@tcu.edu Karel.Netocny@fys.kuleuven.ac.be
1797. ON THE MOST VISITED SITES OF SYMMETRIC MARKOV
PROCESSES
Nathalie Eisenbaum and Davar Khoshnevisan
A growing body of recent works have been devoted to the study
of the favorite points of various concrete Markov processes.
We contribute to this subject by showing that for a large class of
recurrent strongly symmetric Markov processes, singletons are polar
for the most visited site(s).
davar@math.utah.edu
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information provided by the author
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1798. THE DIAMETER OF A LONG RANGE PERCOLATION GRAPH
Don Coppersmith, David Gamarnik, Maxim Sviridenko
We consider the following long range percolation model:
an undirected graph with the node set $Z_N^d$,
has edges $(x,y)$ selected with probability $\beta/||x-y||^s$
if $||x-y||>1$,
and with probability $1$ if $||x-y||=1$, for some parameters $\beta,s>0$.
This model was introduced by Benjamini and Berger, who
obtained bounds on the diameter of this graph for the one-dimensional case $d=1$
and for various values
of $s$, but left cases $s=1,2$ open.
We show that, with high probability,
the diameter of this graph is $\Theta(\log N/\log\log N)$ when $s=d$,
and, for some constants $0<\eta_1<\eta_2<1$,
it is at most $N^{\eta_2}$, when $s=2d$ and is
at least $N^{\eta_1}$
when $d=1,s=2,\beta<1$ or $s>2d$.
dcopper@us.ibm.com gamarnik@watson.ibm.com sviri@us.ibm.com
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